Document Type : Original Article


Laboratoire de Mathematiques et Applications, UFR Mathematiques et Informatique, Uniiversite Felix Houphouet Boigny, Abidjan, Cote d' I voire



 Let $ 1 \leq p\leq \alpha \leq  q \leq \infty$.  The  Fofana's spaces $\left(L^{p},\ell^{q}\right)^{\alpha}(\mathbb{R}^d)$  were introduced in 1988 by Fofana on the basis of Wiener amalgam spaces and their predual spaces $\mathcal{H}(p',q',\alpha')(\mathbb{R}^d)$ have been described by Feichtinger and Feuto in 2019. Recently, in 2023, Yang and Zhou generalized these spaces by replacing the constant exponent $p$ with the variable exponent  $p(\cdot)$ and defining so the variable exponent Fofana's spaces $\left(L^{p(\cdot)},\ell^{q}\right)^{\alpha}(\mathbb{R}^d)$ and their preduals $\mathcal{H}(p'(\cdot),q',\alpha')(\mathbb{R}^d)$.  The purpose of this paper is to investigate the boundedness of classical operators such as Riesz potentials operators, maximal operators, Calderon-Zygmund operators and some generalized sublinear operators in both  $\left(L^{p(\cdot)},\ell^{q}\right)^{\alpha}(\mathbb{R}^d)$ and $\mathcal{H}(p'(\cdot),q',\alpha')(\mathbb{R}^d)$. In order to do this, we prove some properties of these spaces.  Our results extend and/or improve those of classical Fofana's spaces and their preduals.